Cyclic Reduction Research Articles

On a system of difference equations arising in the cyclic reduction

Cyclic reduction is a powerful technique in numerical linear algebra for solving tridiagonal systems. It can be classified in the class of divide and conquer methods. Some of its surprising properties have been studied recently essentially in the case of diagonally dominant matrices. In this paper the case of Toeplitz matrices will be considered. A complete analysis of the properties of the cyclic reduction will be done by studing an associate system of nonlinear difference equations.

Stabilization by perturbation of ILL-conditioned cyclic reduction*

In this work we present a stabilization of the Cyclic Reduction (CR) for tridiagonal systems which consists of a modification of the CR algorithm without back substitution and iterative improving of the results. We consider arbitrary well-conditioned matrices for which we stabilize the algorithm by means of perturbing the results. When it is necessary to divide by numbers which are less than a certain limit δ we perturb them with δ. In consequence, we get a solution which is perturbed, and is close to the exact solution. Then, we apply iterative improvement to obtain a more accurate solution. In practice we need 1-2 iterations to get very accurate solutions. A fast algorithm for the problem with multiple right hand sides is given, as well. At the end we present some numerical evidence including experiments with random matrices.

Increased O2 Consumption and Positive Inotropy Caused by Cyclic GMP Reduction Are Not Altered after L-Type Calcium Channel Blockade

We tested the hypothesis that increased O₂ consumption and inotropy after reduction of myocardial cyclic guanosine monophosphate (cGMP) are mediated through L-type calcium channels. Anesthetized, open-chest New Zealand white rabbits were divided into four groups. Hearts were exposed to control vehicle (n = 8); LY83583 (LY, 10^–3 mol/l, guanylate cyclase inhibitor, n = 9); nifedipine (nif, 10^–4 mol/l, L-type calcium channel blocker, n = 8), or nif+LY (n = 6). Vehicle or compound was applied topically to the epicardium for 15 min. Subepicardial (EPI) blood flow increased (from 213 ± 22 to 323 ± 24 ml/min/100 g) in the presence of LY, as did subendocardial (ENDO) blood flow (from 238 ± 20 to 333 ± 38 ml/min/100 g). O₂ consumption increased in the presence of LY: 18.0 ± 1.0 (EPI) and 17.0±0.6 (ENDO) ml O₂/min/100 g as compared with 9.5 ± 2.0 (EPI) and 10.6 ± 2.5 (ENDO) in the control group. The increase in O₂ consumption with LY was undiminished in the presence of nif (nif+LY group 21.0 ± 3.0 ml O₂/min/100 g EPI and 22.1 ± 3.8 ENDO). Nif alone decreased left ventricular dP/dt_max from (2,762 ± 197 to 2,413 ± 316 mm Hg/s) and maximal rate of change in wall thickness (dW/dt_max from 13.5 ± 2.0 to 9.5 ± 0.8 mm/s), while percent change of wall thickness (from 21.3 ± 3.3 to 31.3 ± 7.2) and dW/dt_max (from 13.3 ± 3.0 to 15.3 ± 2.3 mm/s) increased in the nif+LY group. Thus, the positive O₂ consumption and inotropic effects of decreasing cGMP were undiminished by nif. These results suggest that the cGMP reduction induced increases in O₂ consumption and that inotropy may not be mediated through L-type calcium channels.

Effects of Recombinant Human Megakaryocyte Growth and Development Factor (rHuMGDF) on Platelet Production, Platelet Aggregation, and Thrombosis.

Recombinant human megakaryocyte growth and development factor (rHuMGDF) is a c-mpl ligand that promotes the differentiation of CD34+ precursor cells into megakaryocyte, and then platelets. In experimental animals, injection of this and other c-mpl ligands leads to profound increases in the circulating platelet count in a matter of days. However, c-mpl ligands have also been shown to sensitize platelets to aggregating agents in vitro, raising the possibility that c-mpI ligands may have prothrombotic effects in vivo. Therefore, characterizing rHuMGDF in an in vivo model of thrombosis is a necessary and critical step in defining the in vivo pharmacology of this novel and important hernatopoietic factor, a pegylated form of which is currently in clinical trials. To determine the biologically effective doses in the rabbit, daily subcutaneous injections of rHuMGDF at 0.1, 1.0, or 10 µg/kg were administered ever 7 days. Daily injection of 10 µ/kg produced an approximate fourfold increase in platelet count and 1.0 µ/kg doubled platelet count over the injection period, both of which were statistically significant. The serum concentrations of rHuMGDF were determined 10 minutes following a single intravenous injection with 0.1, 1.0, and 10 µ/kg, and were 0.05 +/- 0.02, 0.98 +/- 0.07, and 21.32 +/- 21.35 ng/ml. To determine whether rHuMGDF can sensitize platelets in vivo, platelet aggregometry was performed on platelets isolated from animals immediately before and 10 minutes after they had been injected intravenously with rHuMGDF (0.1, 1.0, and 10 µ/kg). Intravenous injection of 10 µ/kg produced measurable changes in platelet aggregometry ex vivo, as evidenced by an increased sensitivity of platelets to adenosine diphosphate (ADP). To assess. the in vivo prothrombotic potential of rHuMGDF, a rabbit carotid artery model of cyclic flow reduction (CFR) was used to measure the effect of intravenous rHuMGDF administration on the rate of thrombus formation as assessed by CFR slope and frequency. Intravenous administration of rHuMGDF had no effect on CFR slope or frequency when administered in doses ranging from 0.1 to 10 µ/kg. Control experiments demonstrated that CFR slope and frequency can be enhanced by intravenous infusion of epinephrine and can be abolished by the combined administration of aspirin and ketanserin, indicating that potentially prothrombotic and antithrombotic agents can be identified in this model. We conclude that biologically active doses of rHuMGDF used in this study (1.0 and 10 µ/kg) produce measurable serum levels, induce a thrombopoietic effect, and sensitize platelets in vivo, as determined by ex vivo aggregometry, at 10 µ/kg. Despite the sensitization of platelets to aggregation induced by ADP, it is clear that rHuMGDF does not alter the pattern of CFRs observed in the rabbit carotid artery, whereas agents known to sensitize platelets (epinepherine) and to inhibit platelets (aspirin and ketanserin) readily affected the CFR pattern. These findings indicate that intravenous rHuMIGDF administration, while capable of sensitizing pIatelets, does not enhance platelet-dependent thrombosis in vivo.

Preconditioning for special nonsymmetric systems

AbstractThe paper presents an algorithm for solving nonsymmetric linear systems arising from a two‐cyclic discretization of non‐selfadjoint two‐dimensional elliptic partial differential equations. The algorithm consists of a cyclic reduction repetition using red black ordering. The restarted and preconditioned GMRES method with pre‐iterations is used for solving the final reduced system. Numerical experiments have shown that this procedure accelerates the convergence several‐times.

An Improved Mapping of Cyclic Elimination onto Hypercubes Using Data Replication

In this paper, we propose a new mapping of the Cyclic Elimination (CE) algorithm for the solution of block tridiagonal linear system of equations onto hypercube multiprocessors. Unlike the previous mapping schemes, in our mapping of the CE algorithm all communications are restricted to physically adjacent processors, using the concept of data replication. The effectiveness of our mapping is demonstrated by comparing it with the existing mapping of the Cyclic Reduction algorithm onto hypercubes using both analytical and simulation methods.

Cyclic reduction, dichotomy, and the estimation of differential equations

In the estimation of ordinary differential equations given observed data it is necessary to parametrize the solutions in a computationally tractable way in order to make comparisons with the given data. Typically, this is done by adjoining initial or boundary conditions, and this requires additional information on the solution structure in order to do this in a manner that leads to stable computation of the comparison solutions. Here, an alternative approach is described in which cyclic reduction is used to reduce the estimation problem to an optimization problem subject to a fixed number of equality constraints. If orthogonal transformations are used in the cyclic reduction process then it appears that stable computations are possible without the need for the structural information needed to devise the stable imbeddings. The aim of this paper is to provide evidence in support of this claim. In particular, it is shown that the cyclic reduction process is linked to a new family of representation of the solutions of the ODE system. Some properties of members of this family are described. These provide insight into the particular advantages of the orthogonal reduction form of cyclic reduction.

Reply

Staff Anesthesiologists, Department of General Anesthesiology, The Cleveland Clinic Foundation, 9500 Euclid Avenue, Cleveland, Ohio 44195, Electronic mail: sprungj@cesmtp.ccf.org.In Reply:-Dr. Rosenthal raises two issues concerning our report-alternative treatments and the possibility that the hypotension resulted from acute effects, and not long-term effects, of tricyclic antidepressants (TCAs).We focused our discussion on the controversy of whether direct-acting or indirect-acting sympathomimetics should be given to manage TCA-induced hypotension, an issue of major importance to anesthesiologists dealing acutely with this problem intraoperatively, rather than focusing on the various therapies. We agree with Dr. Rosenthal that the American Heart Association (AHA) guidelines are a mainstay of treatment, and we followed the guidelines; our patient's arterial blood gas pH was 7.47 and the PaCO2was 30 mmHg, thus the pH was in the target range the AHA recommends. Reporting the blood gas values would have provided readers with this information. Instead of using bicarbonate to alkalinize the patient with TCA cardiotoxicity, we induced respiratory alkalosis by hyperventilating the lungs. This approach has the same end-effect-it decreases the free form of the drug in plasma. However, even this approach does not guarantee the success, as was seen in our case and in the case presented by Sener et al. [2] Another treatment for TCA-induced cardiotoxicity is the administration of glucagon (10-mg bolus followed by an infusion of 10 mg over 6 h). [2] The role of glucagon in managing TCA-induced cardiotoxicity is a dose-dependent increase in intracellular cyclic AMP synthesis and reduction of calcium efflux from the cells. [3] Sener et al. [2] found that administration of sodium bicarbonate and volume expansion in their patient with TCA-induced cardiovascular collapse had no effect, whereas administering glucagon immediately normalized the low blood pressure.Although we did not emphasize the possible acute effects of TCAs, we acknowledged that such effects may be important because for the subsequent surgery, our patient had not taken his TCA dose for 24 h and did not experience severe hypotension. However, an acute effect cannot account completely for the cardiotoxicity of the aborted first surgery because, as we stated, the blood pressure and heart rate were still low during the subsequent surgery, never exceeding 110/65 mmHg or 80 beats/min. We then concluded that discontinuing TCAs, even 24 h before surgery, may be beneficial because the plasma TCA concentration has a dose-dependent inhibitory effect on sympathetic nerve activity. [4] Whether this acute effect is a result of TCA centrally inhibiting sympathomimetic actions [4] or blocking the peripheral alpha1receptors [5] is not certain, but we did not ignore the possibility of an acute effect, as Dr. Rosenthal implies.Juraj Sprung, M.D., Ph.D.Peter K. Schoenwald, M.D.Staff Anesthesiologists; Department of General Anesthesiology; The Cleveland Clinic Foundation; 9500 Euclid Avenue; Cleveland, Ohio 44195Electronic mail: sprungj@cesmtp.ccf.org(Accepted for publication July, 8, 1997.)

Redox buffering by melanin and Fe(II) in Cryptococcus neoformans.

Melanin is a fungal extracellular redox buffer which, in principle, can neutralize antimicrobial oxidants generated by immunologic effector cells, but its source of reducing equivalents is not known. We wondered whether Fe(II) generated by the external ferric reductase of fungi might have the physiologic function of reducing fungal melanin and thereby promoting pathogenesis. We observed that exposure of a melanin film electrode to reductants decreased the open-circuit potential (OCP) and reduced the area of a cyclic voltammetric reduction wave whereas exposure to oxidants produced the opposite effects. Exposure to 10, 100, 1,000 or 10,000 microM Fe(II) decreased the OCP of melanin by 0.015, 0.038, 0.100, and 0.120 V, respectively, relative to a silver-silver chloride standard, and decreased the area of the cyclic voltammetric reduction wave by 27, 35, 50, and 83%, respectively. Moreover, exposure to Fe(II) increased the buffering capacity by 44%, while exposure to millimolar dithionite did not increase the buffering capacity. The ratio of the amount of bound iron to the amount of the incremental increase in the following oxidation wave was approximately 1.0, suggesting that bound iron participates in buffering. Light absorption by melanin suspensions was decreased 14% by treatment with Fe(II), consistent with reduction of melanin. Light absorption by suspensions of melanized Cryptococcus neoformans was decreased 1.3% by treatment with Fe(II) (P < 0.05). Cultures of C. neoformans generated between 2 and 160 microM Fe(II) in culture supernatant, depending upon the strain and the conditions [the higher values were achieved by a constitutive ferric reductase mutant in high concentrations of Fe(III)]. We infer that Fe(II) can reduce melanin under physiologic conditions; moreover, it binds to melanin and cooperatively increases redox buffering. The data support a model for physiologic redox cycling of fungal melanin, whereby electrons exported by the yeast to form extracellular Fe(II) maintain the reducing capacity of the extracellular redox buffer.

LIFE PREDICTION FOR ADVANCED FERRITIC STEELS SUBJECT TO THERMAL FATIGUE

Abstract— Thermal fatigue is a well recognised source of damage in headers and steam piping of thermoelectric power plant. This topic has been extensively examined in the past for low alloy ferritic steels typically used in such applications. Experimental evidence obtained in low cycle fatigue testing with tensile hold times on Modified 9Cr1Mo and E911 steels suggests that the Linear Damage Summation rule conventionally used in engineering codes for high temperature damage analysis may not be particulary appropriate for the advanced 9Cr steel family. For this reason two alternatives have been examined: (a) a strain based creep damage evaluation using the R5 ductility exhaustion approach and (b) a creep‐fatigue continuum damage mechanics method. The potential advantages and disadvantages of both are discussed. In addition, results from low cycle fatigue and thermomechanical fatigue tests on crossweld specimens machined from welded joints in the Mod.9Cr1Mo alloy are evaluated. Even if the usual cyclic life reduction factor of 2 with respect to base material behaviour appears adequate to account for the mean trend of cross‐weld results, the large variability observed risks making the use of such a factor non‐conservative for accurate life prediction.

Electrochemical analysis of formation of polynucleotide complexes in solution and at electrode surfaces

The formation of double- and triple-stranded complexes between the synthetic homopolyribonucleotides in solution and at the surfaces of the hanging mercury drop (HMDE) and carbon paste (CPE) electrodes was studied by means of voltammetric and chronopotentiometric methods. It was shown that the cyclic voltammetry reduction signal of poly(C) and the anodic signal of poly(G) obtained with HMDE, as well as the chronopotentiometric oxidation signal of poly(G) obtained with CPE can be used to detect the duplex and triplex formation in solution. In addition to these intrinsic polynucleotide signals, the indicator [Co(phen) 3] 3+ was used to differentiate between single-stranded and ordered structures at CPE. It was shown that the indicator binds preferentially to the duplex poly(G) · poly(C) (formed in solution followed by adsorption at CPE) displaying, however, no preferential binding to the triplex structures. If poly(C) or poly(G) were immobilized at the electrode surface (by adsorption forces) and hybridized with the complementary polynucleotide in solution (at neutral pH), only a small extent of complexation could be detected at HMDE under the given conditions. On the other hand, analogous experiments with CPE suggested formation of ordered polynucleotide structures at the carbon electrode surface. As a result of complexation of poly(C) with poly(G) at neutral pH, formation of more than one ordered structure, including the duplex, was indicated. Polynucleotide complexation performed at pH 3.3 (optimum for formation of the protonated triplex poly(C +) · poly(G) · poly(C) in solution) resulted in the formation of a mixture of structures including triplex and duplex structures. Binding of [Co(phen) 3] 3+ to the product of poly(A) complexation with poly(U) suggested that the yield of the duplex formed at the CPE surface may be influenced by the conformation of the single-stranded polynucleotide anchored at the surface prior to the hybridization. The implications of these findings to DNA biosensor development are discussed.

Sex hormones regulate CFTR in developing fetal rat lung epithelial cells.

Sex hormones modulate two normal processes of late-gestation mammalian lung development: the onset of augmented production of surfactant phospholipids and the loss of mesenchymal cells. As prenatal lung development advances, epithelial chloride secretory pathways diminish as opposing sodium absorptive pathways increase in expression. We hypothesized that sex hormones may influence both the gene expression and functional activity of the chloride channel known as the cystic fibrosis transmembrane conductance regulator (CFTR) in fetal lung epithelium. We report here that sex hormones exert opposite effects on CFTR. Androgen increases and estrogen decreases CFTR functional activity [as assessed by CFTR antisense (but not sense) oligodeoxynucleotide-sensitive adenosine 3',5'-cyclic monophosphate-stimulated cell volume reduction or by glibenclamide-sensitive, amiloride-insensitive transepithelial electrical potential] in primary cultures of fetal rat lung epithelial cells. No alterations in CFTR mRNA levels measured by quantitative polymerase chain reaction amplification of reverse transcripts) accompanied either the changes in functional activity induced by sex hormones or the changes observed during normal development, suggesting that sex hormone modulation of CFTR in antenatal lung occurs at a posttranscriptional level. Our data are consistent with the hypothesis that both androgen and estrogen contribute to the male disadvantage with respect to fetal lung functional development.

Effect of Substituents on the Radical Copolymerization of Ring-Substituted Methyl 2-Cyano-3-Phenyl-2-Propenoates with Styrene

Copolymerizations of ring-substituted methyl 2-cyano-3-phenyl-2-propenoates, RC6H4CH = C(CN)CO2CH3 (R = C2H5, CH3O, Cl, Br, F) with styrene (M 1) were studied in solution in the presence of a radical initiator. Terminal and penultimate kinetic models were applied for best prediction of the copolymer composition. The Alfrey-Price Q and e parameters calculated for the monomers correlate well with the relative reactivity (1/r 1). The relative reactivity showed a tendency to increase with increasing Hammett constant σ, cyclic voltammetric reduction potential, E p, calculated atomic charge, and 13C-NMR chemical shift on the olefinic α-carbon of the ring-substituted monomers.

Effects of metal ions on the substrate-specificity and activity of proton-pumping nicotinamide nucleotide transhydrogenase from Escherichia coli

Nicotinamide nucleotide transhydrogenase catalyzes the reversible reduction of NADP + by NADH and a concomitant proton translocation. It was demonstrated (Glavas, N.A. and Bragg, P.D. (1995) Biochim. Biophys. Acta 1231, 297–303) that the Escherichia coli transhydrogenase also catalyzed a reduction of the NAD-analogue 3-acetylpyridine-NAD + (AcPyAD +) by NADH at low pH and in the absence of (added) NADP(H) and high salt concentrations The mechanism of this reaction has as yet not been explained. In the present study, the E. coli transhydrogenase was purified by affinity chromatography through the NADP(H)-site, rendering the pure enzyme free of NADP(H). Using this preparation it was confirmed that the enzyme readily catalyzes the above reaction. Inhibitors specific for the NADP(H)-site, e.g., palmitoyl-Coenzyme A and adenosine-2′-monophosphate-5′-diphosphoribose, strongly inhibited the reduction of AcPyAD + by NADH, whereas an inhibitor of the NAD(H)-site, adenosine 5′-diphosphoribose, was less inhibitory. This suggests that a lack of metal ions or other ions at low pH induces an unspecific interaction of the NADP(H)-site with AcPyAD + or NADH, presumably NADH, producing a cyclic reduction of AcPyAD + by NADH via NAD(H) bound in the NADP(H) site. A stimulation of reduction of AcPyAD + by NADPH by Mg 2+ present during reconstitution of transhydrogenase in phospholipid vesicles was observed, but it is presently unclear whether this effect is related to that seen with the detergent-dispersed enzyme.© 1997 Elsevier Science B.V. All rights reserved.

Improved cyclic reduction for solving queueing problems

The cyclic reduction technique (Buzbee et al., 1970), rephrased in functional form (Bini and Meini, 1996), provides a numerically stable, quadratically convergent method for solving the matrix equation X = ∑+ ∞ i=0 Xi Ai, where the Ai's are nonnegative k × k matrices such that ∑+ ∞ i=0 Ai is column stochastic. In this paper we propose a further improvement of the above method, based on a point-wise evaluation/interpolation at a suitable set of Fourier points, of the functional relations defining each step of cyclic reduction (Bini and Meini,1996). This new technique allows us to devise an algorithm based on FFT having a lower computational cost and a higher numerical stability. Numerical results and comparisons are provided.

A Cyclic Reduction Approach to the Numerical Solution of Boundary Value ODEs

A parallel algorithm for solving linear systems that arise from the discretization of boundary value ODEs is described. It is a modification of a cyclic reduction algorithm that takes advantage of the almost block diagonal structure of the linear system. A stable modification of the original algorithm is also proposed. Numerical results on a distributed memory parallel computer with 32 processors are presented and discussed.

Structural and electronic effects of alkyl substituents on the main group dithiocarbamato complexes: crystal and molecular structures of tris( N, N′-iminodiethanoldithiocarbamato)metal( III) complexes (M  As, Sb and Bi) and cyclic voltammetric studies on substituted dithiocarbamates

Various dithiocarbamates of As III, Sb III and Bi III with diethyldithiocarbamic acid (dedtc), N-methylaminoethanoldithiocarbamic acid (nmedtc) and N, N′-iminodiethanoldithiocarbamic acid were subjected to cyclic voltammetric reduction studies with a glassy carbon working electrode. One-electron reduction processes occurs around −1.0 V for As and Sb complexes and around −0.5 V for Bi complexes which are completely irreversible. Attachment of the polar ends of the molecule is indicated by the pre- and post-reduction adsorption processes. As the number of −CH 2CH 2OH groups increases, the peak currents increase. In the case of Bi complexes, the reduction process tends to become reversible. In addition, the larger peak current associated with the third response indicates the possible reduction of Bi(dtc) 2(DMF) n . In all the complexes the dtc − ion was lost from M(dtc) 3 − species, with oxidation around +0.6 V in the reverse scans. The crystal structures of As(deadtc) 3 ( 1), Sb(deadtc) 3 ( 2) and Bi(deadtc) 3 ( 3) were determined by single crystal X-ray analysis. The coordination polyhedron in As(deadtc) 3 is a distorted trigonal antiprism, with three strong (2.33–2.35 Å) and three weak (2.82–2.86 Å) AsS bonds. Sb(deadtc) 3 has a coordination polyhedron which can be described as a distorted pentagonal pyramid. One of the SbS bonds is very short (2.46 Å) and differences between others are not significant owing to the larger size of antimony compared with arsenic. The stereochemical requirement of the lone pair on Sb has a more pronounced effect in disturbing the neighbouring atoms than that observed in the corresponding As compounds. The binuclear Bi 2(deadtc) 6 complex is centrosymmetric, two bidentate ligands being bridged in such a way that each sulfur atom is simultaneously bonded to both metal ions. The resulting coordination polyhedron can be described as a distorted square antiprism. The asymmetry in the MS (M  As, Sb, Bi) bonds is a consequence of the stereochemical requirement for the ligand and the lone pair of electrons on the metals. This is supported by the valence bond sums (VBS) calculated for the complexes.

Stability of the block cyclic reduction

The forward stability of the block cyclic reduction without back substitution for block tridiagonal systems is studied. The basic assumption is that the matrix of the system is block column diagonally dominant. Then it is shown that for nonstrictly diagonally dominant matrices the forward error is O( C n n 2log 2 nκϱ 0), and for strictly diagonally dominant matrices it is O( C n g( s) log 2 nκϱ 0), where n is the block size of the matrix, N is the size of each block, g( s) is a function which measures the diagonal dominance, κ is a condition number, and ϱ 0 is the machine roundoff unit. At the end some numerical evidence is presented.

On the Solution of a Nonlinear Matrix Equation Arising in Queueing Problems

By extending the cyclic reduction technique to infinite block matrices we devise a new algorithm for computing the solution $G_0 $ of the matrix equation $G = \Sigma _{i = 0}^{ + \infty } G^i A_i $ arising in a wide class of queueing problems. Here $A_i ,i = 0,1, \cdots ,$ are $k \times k$ nonnegative matrices such that $\Sigma _{i = 0}^{ + \infty } A_i $ is column stochastic. Our algorithm, which under mild conditions generates a sequence of matrices converging quadratically to $G_0 $, can be fully described in terms of simple operations between matrix power series, i.e., power series in z having matrix coefficients. Such operations, like multiplication and reciprocation modulo $z^m $, can be quickly computed by means of FFT-based fast polynomial arithmetic; here m is the degree where the power series are numerically cut off in order to reduce them to polynomials. These facts lead to a dramatic reduction of the complexity of solving the given matrix equation; in fact, $O( k^3 m + k^2 m \log m )$ arithmetic operations are sufficient to carry out each iteration of the algorithm. Numerical experiments and comparisons performed with the customary techniques show the effectiveness of our algorithm. For a problem arising from the modelling of metropolitan networks, our algorithm was about 30 times faster than the algorithms customarily used in the applications. Cyclic reduction applied to quasi-birth-death (QBD) problems, i.e., problems where $A_i = O$ for $i > 2$, leads to an algorithm similar to the one of [Latouche and Ramaswami, J. Appi. Probab., 30 (1993), pp. 650–674], but which has a lower computational cost.

The formation of bilayered nickel-iron, cadmium-iron and cadmium—silver hexacyanoferrates by an electrochemically driven insertion—substitution mechanism

Upon cyclic oxidation and reduction of a parent metal hexacyanoferrate, metal ions from the electrolyte solution can be driven into interstitial positions of the zeolitic structure of the metal hexacyanoferrate. Whereas alkali metal ions are stable on these sites, some other metal ions undergo a substitution reaction with the N-coordinated metal ions, since the latter are only weakly bonded. This results in the formation of a layer of a daughter metal hexacyanoferrate covering the parent metal hexacyanoferrate. A bilayered structure is formed which can be identified by its behaviour in cyclic voltammetry. There is no indication that the electron and ion transfer through the interface of the two layers is hindered. From this and the growth mechanism it is probable that the daughter hexacyanoferrate will form an epitaxial layer on top of the parent hexacyanoferrate, provided that the two hexacyanoferrates do not differ too much in their lattice parameters. Experimental results, for which abrasive stripping voltammetry was used, are reported for a nickel hexacyanoferrate layer on Prussian blue, for a cadmium hexacyanoferrate layer on Prussian blue, and for a cadmium hexacyanoferrate layer on silver hexacyanoferrate.